Limits of agreement (Bland-Altman method)

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Statement b is true, whereas a, c, and d are false.

The aim of the study was to assess the extent of agreement between primary care and daytime ambulatory monitoring in the measurement of blood pressure. Evidence was needed from primary care of the implications of using ambulatory monitoring, both in the initiation of treatment and the monitoring of hypertension control. Blood pressure recorded in primary care and by daytime ambulatory monitoring is unlikely to be exactly the same for all patients. If the differences were small, perhaps daytime ambulatory monitoring could replace measurement in primary care or the two methods could be used interchangeably. Although systolic and diastolic blood pressures were investigated separately, only the results for systolic blood pressure are discussed here.

The researchers presented a scatter plot of systolic blood pressure measurements in primary care against those obtained by daytime ambulatory monitoring (fig 2⇓). There would have been perfect agreement between the two methods if all points in the scatter plot were on the line of equality. The line of equality is the straight line shown, with all points on the line having the same value of systolic blood pressure for both methods. Often agreement between two methods of measurement is inappropriately assessed using correlation. There was a significant correlation between the two measurements of systolic blood pressure (Spearman’s r=0.46, P<0.05) providing evidence of a linear association. However, this is not evidence that the two methods of measurement agree (a is false). Spearman’s correlation coefficient has been described in a previous question.2 The correlation measured the strength of the linear association between the two measurements of systolic blood pressure. Perfect correlation would exist if all the points in the scatter plot were on any straight line (providing it was not horizontal) and not only for the line of equality. If ambulatory daytime monitoring gave a systolic blood pressure measurement that was half of that obtained in primary care, for example, then the two methods would show no agreement yet perfect correlation.

Fig 2 Scatter plot of systolic blood pressure measured by general practitioner against daytime ambulatory systolic pressure, with the line of equality indicated

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Bland and Altman suggested an approach to investigating the extent of agreement between two methods of measurement based on graphical techniques and straightforward calculations.3 The Bland-Altman plot (fig 1) charts the difference in systolic blood pressure measurements (primary care minus daytime ambulatory monitoring) on the vertical axis against the average of the two measurements. The average of the two measurements for each patient is considered a better estimate of the true systolic blood pressure than either measurement alone. Therefore, the Bland-Altman plot enables a visual inspection of the association between the difference in measurements and the magnitude of systolic blood pressure.

Four horizontal lines are displayed on the Bland-Altman plot (fig 1). The thin black line crossing the vertical axis at zero represents no difference between primary care and daytime ambulatory monitoring in systolic blood pressure measurement. In this instance, most of the points were above the line of no difference, indicating that measurements of systolic blood pressure in primary care were greater than for daytime ambulatory monitoring for most patients. The measurements in primary care exceeded those of daytime ambulatory monitoring by an average of 18.9 mm Hg, as indicated by the thick blue line on the Bland-Altman plot. The standard deviation of the differences (GP minus ambulatory daytime monitoring) in systolic blood pressure was 19.0 mm Hg. For about 95% of patients, their difference in systolic blood pressure will be within two standard deviations of the mean difference (b is true). This interval is (18.9−2(19)) to (18.9+2(19)) mm Hg, which equals −19.1 to 56.9 mm Hg, and is represented by the broken red lines on the plot. The limits of this interval are called the limits of agreement. The differences in systolic blood pressure do not have to follow a normal distribution for the limits of agreement to be calculated (c is false). A previous question described how the sample standard deviation can be used to calculate a series of ranges, containing certain percentages of the sample members, to describe the variation in the measurements of a variable regardless of the distribution of the measurements.4

If the points on the Bland-Altman plot were uniformly scattered between the limits of agreement, it might suggest good agreement between the two methods of measurement. About 95% of patients would have a difference in systolic blood pressure measurements (primary care minus daytime ambulatory monitoring) within the limits of agreement—that is, in the −19.1 to 56.9 mm Hg interval. This interval is wide, indicating that large differences in measurements were seen. However, it is a matter of clinical judgment how far apart measurements can be before two methods can no longer be considered interchangeable. The measurements made by the doctors exceeded those obtained by daytime ambulatory monitoring by an average of 18.9 mm Hg. The researchers discussed whether 18.9 mm Hg could be added to the daytime ambulatory systolic measurements to provide “equivalent” clinic readings to inform decision making. It was suggested that the observed differences in systolic blood pressure measurements were partly caused by the white coat effect, whereby blood pressure is raised in the clinical setting because of anxiety induced by the clinic visit.

The Bland-Altman plot suggests an association between the difference in systolic pressure and magnitude of systolic blood pressure; in particular, the difference increases as systolic blood pressure increases. This is also suggested by the significant correlation between the points on the Bland-Altman plot (Spearman’s r=0.32; P<0.05), which provides evidence of a linear association. Some large differences were seen when systolic pressure was high. Therefore, poor agreement exists between primary care and daytime ambulatory monitoring in the measurement of systolic blood pressure (d is false). Logarithmic transformation, described in a previous question,5 of the systolic blood pressure measurements might remove the association between the difference in systolic pressure measurements and increasing mean measurement. To do this a Bland-Altman plot should be constructed for the log transformed data, with the corresponding limits of agreement obtained as described above. If there was no association between the difference in measurements and the mean measurements when the data were log transformed, it would indicate that there was good agreement. However, the limits of agreement on the log scale would need to be related to the original scale of measurement. Therefore, the antilogs of the limits should be obtained to describe the variation in the difference between measurements on the original measurement scale.